IPMAT Indore 2022 Question Paper with Answers

For $0\lt\theta\lt\frac{\pi}{4}$, let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right), c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$. Then, the median value in the sequence $a, b, c, d$ is

In a room, there are $n$ persons whose average height is $160 \ \text{cm}$. If $m$ more persons, whose average height is $172 \ \text{cm}$, enter the room, then the average height of all persons in the room becomes $164 \ \text{cm}$. Then $m: n$ is

The sum of the first 15 terms in an arithmetic progression is 200, while the sum of the next 15 terms is 350. Then the common difference is

Suppose $a, b$ and $c$ are integers such that $a>b>c>0$, and $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$. Then the determinant of $A$

In a right-angled triangle ABC, the hypotenuse AC is of length 13 cm. A line drawn connecting the midpoints D and E of sides AB and AC is found to be 6 cm in length. The length of BC is

If $A=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 0\end{array}\right]$ then $A^{2022}$ is

Let $A=\{1,2,3\}$ and $B=\{a, b\}$. Assuming all relations from set $A$ to set $B$ are equally likely, what is the probability that a relation from $A$ to $B$ is also a function?

In a 400-metre race, Ashok beats Bipin and Chandan respectively by 15 seconds and 25 seconds. If Ashok beats Bipin by 150 metres, by how many metres does Bipin beat Chandan in the race?

In a bowl containing 60 ml orange juice, 40 ml of water is poured. Thereafter, 100 ml of apple juice is poured to make a fruit punch. Madhu drinks 50 ml of this fruit punch and comments that the proportion of orange juice needs to be higher for better taste. How much orange juice should be poured into the fruit punch that remained, in order to bring up the level of orange juice to 50 percent?

The curve represented by the equation $\dfrac{x^{2}}{\sin \sqrt{2}-\sin \sqrt{3}}+\dfrac{y^{2}}{\cos \sqrt{2}-\cos \sqrt{3}}=1$ is

A set of all possible values the function $f(x)=\dfrac{x}{|x|}$, where $x \neq 0$, takes is

The set of real values of $x$ for which the inequality $\log _{27} 8 \leq \log _{3} x \lt 9^{\frac{1}{\log _{2} 3}}$ holds is

When the square of the difference of two natural numbers is subtracted from the square of the sum of the same two numbers and the result is divided by four, we get

The cost of a piece of jewellery is proportional to the square of its weight. A piece of jewellery weighing 10 grams is INR 3600. The cost of a piece of jewellery of the same kind weighing 4 grams is

Let $A$ and $B$ be two sets such that the Cartesian product $A \times B$ consists of four elements. If two elements of $A \times B$ are $(1,4)$ and $(4,1)$, then

If $f\left(x^{2}+f(y)\right)=x f(x)+y$ for all non-negative integers $x$ and $y$, then the value of $[f(0)]^{2}+f(0)$ equals _________.

If one of the factors of the number $3^{7} 2^{8} 17^{3}$ is randomly chosen, then the probability that the chosen factor will be a perfect square is

The number of four-digit integers which are greater than 1000 and divisible by both 2 and 3, but not by 5, is

A showroom is open on all seven days of the week throughout the year. There are five employees Alex, Bhabha, Cathy, Dilip and Ethan who work in the showroom. Every day except Sunday, two employees are required while on Sunday three employees need to work. Every employee works for three days in a week. Some additional information is also provided:

- Every employee works on two consecutive days while the third day is not consecutive to either of them. (The week is continuous, so Sunday and the following Monday count as consecutive days.)
- Alex and Dilip work together on Tuesday and Wednesday while the other working day differs for them.
- Neither Bhabha nor Cathy works with Alex on any day.
- Cathy does not work either on Saturday or on Monday.

Number of days Bhabha and Cathy work together in a week is

A showroom is open on all seven days of the week throughout the year. There are five employees Alex, Bhabha, Cathy, Dilip and Ethan who work in the showroom. Every day except Sunday, two employees are required while on Sunday three employees need to work. Every employee works for three days in a week. Some additional information is also provided:

- Every employee works on two consecutive days while the third day is not consecutive to either of them. (The week is continuous, so Sunday and the following Monday count as consecutive days.)
- Alex and Dilip work together on Tuesday and Wednesday while the other working day differs for them.
- Neither Bhabha nor Cathy works with Alex on any day.
- Cathy does not work either on Saturday or on Monday.

Which among the following employees do not work together on any of the days?